Lifting of homeomorphisms to 4-sheeted branched coverings of a disk
type: Item
instance of: scholarly article
title: Lifting of homeomorphisms to 4-sheeted branched coverings of a disk
zbMATH Open document ID: 1154.57002
author: Agnieszka Wisniowska-Wajnryb
author: Bronisław Wajnryb
published in: Topology and its Applications
publication date: 2008-09-23T00:00:00Z
review text: Let \(p:X\to D\) be a 4-sheeted branched covering of a closed disk \(D\), possibly not connected, with \(n\) branch values \(A_1\), \(A_2,\ldots,A_n\in D^\circ\). It is supposed that \(p\) is simple, i.e. for each \(A_j\) the preimage \(p^{-1}(A_j)\) consists of \(3\) points. Each loop \(\sigma\) at a fixed point \(A_0\in\partial D\), lying in \(D\setminus\{A_1\), \(A_2,\ldots,A_n\}\), defines a permutaion of leaves of the branched covering, so we have the monodromy homeomorphism \(\mu:\pi_1(D\setminus\{A_1\), \(A_2,\ldots,A_n\},A_0)\to\Sigma_4\); here \(\Sigma_4\) is the symmetric group of order~\(4\). We fix a sequence of \(n\) disjoint (except for the initial points) simple curves \(\alpha_1\), \(\alpha_2,\ldots,\alpha_n\); each \(\alpha_j\) connects \(A_0\) to \(A_j\) in \(D\) and does not meet \(A_i\), \(i\neq j\). Let the curves \(\alpha_1\), \(\alpha_2,\ldots,\alpha_n\) meet at \(A_0\) in the clockwise order. For each \(j\) denote by \(\widehat{\alpha}_j\) the loop which goes along \(\alpha_j\) to a sufficiently small neighbourhood \(U_j\) of \(A_j\) then turns clockwise around \(A_j\) in \(U_j\) and comes back to \(A_0\) along \(\alpha_j\). Denote by \(\mu(\alpha_j)\) the image of the homotopy class of \(\widehat{\alpha}_j\) under the monodromy homeomorphism. By definition, the sequence \(\mu(\alpha_1)\), \(\mu(\alpha_2),\dots,\mu(\alpha_n)\) is a monodromy sequence for \(p\). Now we remember the concept of half-twist. Let \(x\) be a simple arc in \(D^\circ\) connecting two distinct branch values and free of the others. Consider a sufficiently small neighbourhood of \(x\). It is homeomorphic to the closed unit disk \(U\) with \(x\) corresponding to a segment centered at the origin. The half-twist around \(x\) is the isotopy class of the homeomorphism induced by the following homeomorphism \(h\) of \(U\). The map \(h\) is obtained from the identity map by rotation of \(y\) around the origin by \(180^\circ\) with \(\partial U\) pointwise fixed. The isotopy cla…
Mathematics Subject Classification ID: 57M12
zbMATH DE Number: 5346285
zbMATH Keywords: branched covering
zbMATH Keywords: surface
zbMATH Keywords: monodromy group
zbMATH Keywords: braid group
reviewed by: Samyon R. Nasyrov
MaRDI profile type: Publication
full work available at URL: j.topol.2008.06.002
OpenAlex ID: W1994496591
cites work: On \(4\)-fold covering moves
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DOI: 10.1016/J.TOPOL.2008.06.002
description: scientific article; zbMATH DE number 5346285